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Problem 8 Easy Difficulty

If $ a = i - 2k $ and $ b = j + k $, find $ a \times b $. Sketch $ a, b $, and $ a \times b $ as vectors starting at the origin.

Answer

$\mathbf{a} \times \mathbf{b}=2 \mathbf{i}-\mathbf{j}+\mathbf{k}$

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Video Transcript

Let's try another cross product problem. Where a. is the vector one, I plus zero, J minus two K. And B. Is the vector zero, I plus one. J plus one K. And we can mark those that are matrix here. And using the technique from our textbook to find the cross product A cross B. We can ignore the first column and look at zero times 1 -20 times one minus negative two times one. All multiplied by I minus. Then we'll ignore our second column. And we look at one times one minus negative two times zero. One times one minus negative too Time zero. All Times J. And lastly will ignore our 3rd column And look at one times 1 zero times 0. That's one zero times 0. Okay, if we simplify all of this, we have zero times one is zero minus negative, two times one is negative two. So zero minus negative too is zero plus two. I uh huh minus one times one -2 times zero jay plus one times one zero times 0. Okay. Yeah, if we want to write that as a vector Then we can just simplify all of this down to two negative one, one. What happens if we want to plot all of these? Let's say we want to sketch a. B. And this new vector. This cross product. Let's call it C. In three D. Space. Well, we'll need to start with a set of axes. We've got our X axis coming under the screen. Why access running horizontal? And does that access going vertical? I apologies, apologized to the Canadian nests. If we want to plot a That's 10 -2 or one step in the X direction. Zero steps in the Y. Direction And the -2 steps in the Z direction. So 10 and then down to Mhm. We'll call this one hey and we can write a line or an arrow representing our vector straight up to that point. If we want to try b. steps in the extraction, so your steps this way one step in the Y. Direction And one step vertically. So complaint right around here and label it big. And lastly our cross product A cross B. Is two steps in the x. direction, 12 negative one in the Y. Direction and one vertically. If we go over out to Over one up, 1 create something like this and we can label it a cross B. And if we did this right, This should be perpendicular to both of those, which is a little difficult to visualize in three D. But that's our answer.